p −1-Linear Maps in Algebra and Geometry

Abstract

In this article we survey the basic properties of p −e -linear endomorphisms of coherent OX -modules, i.e. of OX -linear maps F * F − → G where F , G are OX -modules and F is the Frobenius of a variety of finite type over a perfect field of characteristic p > 0. We emphasize their relevance to commutative algebra, local cohomology and the theory of test ideals on the one hand, and global geometric applications to vanishing theorems and lifting of sections on the other. p −1 -LINEAR MAPS IN ALGEBRA AND GEOMETRY 5 Exercise 2.7. Suppose that X is a projective variety with ample line bundle L and suppose that F is a coherent sheaf on X. Set S = i∈Z H 0 (X, L i ) to be the section ring with respect to L and set M = i∈Z H 0 (X, F ⊗

A new subadditivity formula for test ideals

“…In this section, we provide the basic definitions and results from the theory of test ideals and Cartier algebras. A more complete account of these theories may be found in the surveys, [ST11,BS13].…”

“…Recall that for any normal F -finite scheme 6 X of characteristic p and any map ϕ : F e O X Ñ O X we can associate an effective Q-divisor ∆ ϕ on X such that K X`∆ϕ is Q-Cartier with Cartier index not divisible by p (Cf. [BS13], [Sch09, Section 3]). If h is a global section of O X and we set ϕ h " ϕpF e h¨´q, then ∆ ϕ h " ∆ ϕ`1 {pp e´1 q div h. Now suppose that X is an F -finite integral scheme with fraction field sheaf K pXq.…”

The TestIdeals package for Macaulay2

“…This latter description is not computable, however. In the case that R is Gorenstein, Hom R (R 1/p e , R) is a cyclic R 1/p e -module generated by Φ e , which corresponds with the map T above based on the identification ω R ∼ = R [BS13]. More generally, if R is Q-Gorenstein with index not divisible by p, then for at least sufficiently divisible e > 0, such a generating Φ e still exists.…”

A Koszul complex over skew polynomial rings

“…More generally, we may consider the e-th powers ϕ e of the ring homomorphism ϕ and define the corresponding notion of ϕ e and ϕ −e -linear maps. We may collect all these morphisms in a suitable algebra that would provide a generalization of the Frobenius algebra introduced by G. Lyubeznik and K. E. Smith in [LS01, Definition 3.5] and the Cartier algebra considered by K. Schwede [Sch11] and generalized by M. Blickle [Bli13] (see also [BS13]).…”